Compressive sensing is a well-known theory that has been widely published in the geophysical community since at least 2008. Compressive sensing is used in connection with seismic data acquisition and seismic data reconstruction to ultimately image the seismic data for purposes of locating hydrocarbon resources. In the field of seismic data acquisition, compressive sensing applies the well-known concept of non-uniform sampling to determine optimal shot point (hereinafter referred to as source) and sensor (hereinafter referred to as receiver) locations in respective pre-plot seismic survey designs that are used for acquiring seismic data. Non-uniform sampling exploits the sparsity of the signals sampled to optimally recover information about the formation from far fewer samples than those required by more conventional sampling techniques following the Nyquist-Shannon sampling theorem. The seismic data is non-uniformly sampled (acquired) so that normally coherent aliased energy is incoherently sampled while maintaining the coherent sampling of the desired seismic data. Compressive sensing therefore, requires a sparse representation of the seismic signals sampled in the transform domain and incoherent noise in the transform domain. Once the seismic data is acquired, it may be used with conventional compressive sensing techniques to reconstruct seismic data in unsampled locations.
Because compressive sensing can be used to reduce the number of sources and receivers required to acquire the seismic data and/or improve the resolution of a seismic data image, different techniques have been proposed to maintain a sparse representation of the seismic signals sampled and incoherent noise. One technique utilizes mutual coherence and randomized greedy algorithms, both well-known concepts, to determine an optimal sampling grid for source and receiver locations. These concepts are used together to determine an optimal sampling grid for source and receiver locations when a mutual coherence value is minimized for the optimal sampling grid. Using a single, minimized, mutual coherence value to evaluate the level of incoherence, however, may oversimplify optimization for complex surveys and lead to less accurate seismic data reconstruction results using compressive sensing techniques. Other techniques utilize misaligned sources and/or receivers to maintain a sparse (random) representation of the seismic signals sampled and incoherent noise. Such techniques, however, limit the transform domain and require local interpolation to reconstruct the seismic data in unsampled locations thus, producing less accurate seismic data reconstruction results using compressive sensing techniques.